Game-based Pricing and Task Offloading in Mobile Edge Computing Enabled Edge-Cloud Systems
11 Game-based Pricing and Task Offloading in MobileEdge Computing Enabled Edge-Cloud Systems
Yi Su, Wenhao Fan,
Member, IEEE , Yuan’an Liu,
Member, IEEE , and Fan Wu
Abstract —As a momentous enabling of the Internet of things(IoT), mobile edge computing (MEC) provides IoT mobile devices(MD) with powerful external computing and storage resources.However, a mechanism addressing distributed task offloading andprice competition for the open exchange marketplace has notbeen established properly, which has become a huge obstacle toMEC’s application in the IoT market. In this paper, we formulatea distributed mechanism to analyze the interaction between OSPsand IoT MDs in the MEC enabled edge-cloud system by applingmulti-leader multi-follower two-tier Stackelberg game theory. Wefirst prove the existence of the Stackelberg equilibrium, andthen we propose two distributed algorithms, namely iterativeproximal offloading algorithm (IPOA) and iterative Stackelberggame pricing algorithm (ISPA). The IPOA solves the followernon-cooperative game among IoT MDs and ISPA uses backwardinduction to deal with the price competition among OSPs.Experimental results show that IPOA can markedly reduce thedisutility of IoT MDs compared with other traditional taskoffloading schemes and the price of anarchy is always less than150%. Besides, results also demonstrate that ISPA is reliable inboosting the revenue of OSPs.
Index Terms —Internet of things, mobile edge computing,Stackelberg game
I. I
NTRODUCTION W ITH the rapid development and popularization ofInternet of Things (IoT) technology, the number ofdeployed IoT devices is experiencing explosive growth [1].It is estimated that the number of IoT devices will reachan astonishing 21.5 billion by 2025 [2]. These devices willgenerate massive data and demand further processing, pro-viding intelligence to both service providers and users [3].However, most IoT devices, especially mobile devices (MDs)have highly constrained computing power and battery capacity,which means that it is unrealistic to meet the demands of IoTapplications by processing all the raw data locally. Althoughtraditional cloud computing allows IoT MDs to offload theircomputation tasks to the remote data center so as to utilizepowerful central processing unit (CPU) and ample storagecapabilities of cloud [4], considerable transmission costs areincurred because the data centers that cloud computing relieson are geographically far away from MDs. In addition to this,offloading tasks to the cloud imposes huge additional burdenson the backbone network traffic load and the performance ofthe network will worsen with increasing data size.In order to overcome the challenges associated with central-ized cloud computing environments, the European Telecom-munications Standards Institute (ETSI) introduced the term
Yi Su, Wenhao Fan, Yuan’an Liu and Fan Wu are with the School ofElectronic Engineering, and Beijing Key Laboratory of Work Safety IntelligentMonitoring, Beijing University of Posts and Telecommunications, Beijing,China (e-mail: [email protected], [email protected]).
Mobile Edge Computing (MEC) in 2014 and further droppedthe ’Mobile’ out of MEC and renamed it as Multi-accessEdge Computing in order to broaden its applicability intoheterogeneous networks in 2016 [5]. MEC is initiated aimingto create an open environment across multi-vendor cloudplatforms located at the edge of access network, accessibleby application/service providers and third parties [6]. Seenas a key technology for 5G wireless systems, the servers ofMEC can be located at the base stations (BSs) in a fullydistributed manner, enabling the delivery of locally-relevant,fast services. Offloading tasks to the MEC servers at BSswill bring markedly diminishing transmission delay comparedwith mobile cloud offloading. With the help of MEC, sensitivedata can also be confined to local zones and not exposed tothe internet, enhancing IoT information security. As lots ofliterature advocate [7], [8], [9], MEC is not a substitute but acomplement to cloud computing and a cooperative edge-cloudsystem can provide IoT MDs with diverse offloading choicesto improve their quality of experience (QoE).Although mobile computation offloading in MEC has beenextensively studied in the literature [8], [9], [10], [11], [12],there are still many challenging issues with integrating MECto assist IoT: (i) IoT MDs are more sensitive to delay andenergy consumption and the computation tasks of differentIoT MDs vary greatly, all of which make it difficult to modelthe utility function of IoT MDs; (ii) Due to IoT connectinga diverse assortment of devices belonging to different parties,the service resources may not belong to a single offloadingservice provider (OSPs) [3]; (iii) Selfish OSPs and IoT MDsare interested in optimizing their own utility individuallyin the collaborative edge-cloud MEC system, which furtherincreases the difficulty of offloading strategy optimization andsystem stability. Many studies leverage auction theory to as-sign computing resources and design pricing policies betweenmultiple OSPs and IoT MDs [13], [14], [15]. However, in thosestudies, there is always a trustworthy third party acting as anauctioneer, which may not be found in some IoT scenarios.Therefore, designing a distributed computing offloading andpricing mechanism in such an authority-lacking competitiveedge-cloud system is still an urgent problem to be solved,which interests us in conducting an in-depth study of thisproblem.In this paper, we are concentrating on designing a noveldistributed task offloading and pricing mechanism in anedge-cloud enabled IoT environment, where multiple edgeor cloud OSPs provide distinct offloading services and IoTMDs will offload their computation tasks to different edgeservers and cloud servers proportionally, according to theprices announced by OSPs. To address the urgent problem a r X i v : . [ c s . G T ] J a n of task offloading and pricing in a fully distributed manner,we place the trading between OSPs and IoT MDs in an openexchange marketplace and the mechanism based on economicprinciples is applied in such marketplace. OSPs are seen asleaders and IoT MDs are considered as followers becausethe offloading strategies of IoT MDs are determined afterOSPs’ prices are given. The interaction between OSPs and IoTMDs is formulated as a multi-leader multi-follower two-tierStackelberg game [16] and Stackelberg equilibrium (SE) existsin our proposed mechanism. Our main contributions include:1) Modeling . A novel disutility function is formulatedfor IoT MDs to quantify their QoE. Comprehensivelyconsidering queuing models at different stages and theindividual differences in the importance of delay, energyconsumption and payment, this model can accuratelyreflect how the pricing of OSPs and the offloadingstrategies of IoT MDs affect the QoE of IoT MDs.2)
Game analysis . The interaction between OSPs and IoTMDs is regarded as a multi-leader multi-follower two-tier Stackelberg game, which is composed of a leadernon-cooperative game for OSPs and a follower non-cooperative game for IoT MDs. We employ the varia-tional inequality (VI) approach to analyze the existenceand uniqueness of Nash equilibrium (NE) in the followernon-cooperative game given the prices of OSPs. Forthe leader non-cooperative game, the existence of NEis also derived, verifying the existence of Stackelbergequilibrium (SE).3)
Algorithms design . A distributed iterative proximal of-floading algorithm (IPOA) is proposed to address theproblem of task offloading given the prices of all OSPs.This algorithm can converge to an NE in limited itera-tions. Additionally, we propose another iteritive Stackel-berg game pricing algorithm (ISPA) to solve the leadernon-cooperative pricing game among OSPs by applyingbackward induction.4)
Performance evaluation . Simulations are conducted toinvestigate the performance of our proposed mecha-nism. Results show that IPOA can markedly improveIoT MDs’ QoE compared with other traditional taskoffloading schemes. The price of anarchy (PoA), whichreflects the gap in overall performance between NE andsocially optimal offloading, is bounded. We further studythe ISPA and prove that OSPs can find appropriate pricesfor all OSPs.The rest of the paper is organized as follows. In Sec. II, weintroduce related works. The system model is described in Sec.III. Based on the system model, we analyze the Stackelberggame between OSPs and IoT MDs in Sec. IV. Furthermore, inSec. V, two algorithms are proposed to deal with distributedtask offloading and pricing problems. Sec. VI investigates theperformance of our proposed algorithms. We conclude thepaper in Sec. VII. II. R
ELATED W ORKS
The computation offloading, and partial offloading in par-ticular, is a very complex process affected by different factors, such as users preferences, radio and backhaul connectionquality and cloud capabilities [4]. Considering different factorsor adopting different methods, many literatures have launchedstudies into mobile computing offloading in MEC. Fan et al.[12] considered the problem of excessive load on a specificMEC-BS, and the scheme they proposed could not optimizethe performance of the entire system. In [9], the authorsinvestigated the performance gains of cooperative edge-cloudcomputation offloading for MEC enabled FiWi enhanced Het-Nets and presented a self-organization based mechanism toenable mobile users. Liu et all. [17] brought a thorough studyon the energy consumption, execution delay and payment costof offloading processes in a fog computing system whereonly one OSP exists. They proposed an algorithm aiming tominimize the average computing cost of all MDs in a centralmanner, which did not apply to competitive environments.The authors of [18] considered the task queue state, theenergy queue state as well as the channel qualities betweenMU and BSs in the MEC systems and modeled the optimalcomputation offloading problem as a Markov decision process.Deep reinforcement learning based algorithms were proposedto optimize task computation experience of mobile users.Although there are many approaches using deep reinforcementlearning to design computation offloading policies in theMEC, even some works like [19] applies deep reinforcenmentlearning to solve the competition among OSPs, we have not yetseen the possibility of applying deep reinforcement learningbased optimization methods to address computation offloadingand pricing jointly in distributed edge-cloud systems.As a well-researched economic theory, auction theory hasbeen employed by many literatures to allocate resources fromedges to devices in non-cooperative scenarios. In [13], Jin et al.proposed an incentive-compatible auction mechanism (ICAM)based on the single-round double auction model for thecloudlet scenario. The ICAM can effectively allocate cloudlets(sellers) to satisfy the service demands of mobile devices(buyers) and determine the pricing. Wang et al. designedan online profit maximization multi-round auction (PMMRA)mechanism for the computational resource trading betweenedge clouds and mobile devices in a competitive MEC environ-ment and an outperforming profit of edge clouds was obtainedin [14]. In [15], energy harvesting-enabled MDs were consid-ered as offloading service providers and the offloaded tasksgenerated by IoT devices were optimally assigned through aproposed online rewards-optimal auction (RoA). Despite thefocus on the resource allocation on the provider side, thesestudies ignore the impact of MD’s offloading strategy.Stackelberg game model is a classic model in game theoryand several works advocate Stackelberg game theory as aneffective solution concept for pricing and resource allocat-ing problems in the game between resource providers andconsumers [20], [21], [22], [23]. In [22], the complicatedinteractions among unmanned aerial vehicles (UAVs) andBSs as well as the cyclic dependency was considered asa Stackelberg game where BSs were leaders determiningthe bandwidth allocated to each UAV and UAVs acted asfollowers to select the bandwidth. The authors solved theStackelberg game through backward induction as the UAV
Base Station
MD MD MD
EdgeServer EdgeServer EdgeServerCloudServers CloudServers
Fig. 1: System architecturepayoff information was requested by BSs. Jie et al. [23]utilized the double-stage Stackelberg game to propose anoptimal resource allocation scheme between cloud center (CC)and data users (DUs) by introducing fog service providers(FSPs) for a fog-based industrial internet of things (IIoT)environment. They modelled the competition of FSPs as a non-cooperative game in respect of the fact that FSPs competed tobuy resources from the CC in order to provide paid services toDUs. However, they did not consider the competition betweenDUs assuming the resources needed by each DU are distinct.Three algorithms were designed and Nash equilibrium andStackelberg equilibrium were achieved in the end.To the best of our knowledge, there are still no solutions toaddress task offloading and pricing jointly in a fully distributedmanner for the competitive heterogeneous edge-cloud systems,where not only the utility information of each IoT MD isprivate but also the pricing of an OSP will not be known inadvance by other OSPs before it is released. Accurately, how toachieve the equilibrium of such a system is still a challengingproblem. III. S
YSTEM M ODELS
As Fig. 1 depicts, we consider there is a set of M IoT MDs,which is denoted as M = { , ..., M } , within the coverage of aBS and it is assumed that the M IoT MDs can can only accessthe same BS at the same time. All of these IoT MDs can accessto the BS simultaneously. There are N OSPs in the system,which consist of N e edge computing OSPs and N c cloudcomputing OSPs. Each edge computing OSP deploys an edgeserver at the BS and each cloud computing OSP owns a cloudcomputation center connected to BS via optical backbonenetwork [9]. We define the set of OSPs as N = { , ..., N } , ofwhich the first N c elements represent cloud computing OSPsand the rest are edge computing OSPs. The prices announcedby OSPs are broadcast via BS periodically. After receiving theprices, IoT MDs determine their offloading strategies and sendtheir offloading requests to the BS. Compared with task inputdata, the communication data used for publishing prices andnotifying offloading requests is much smaller, which ensuresthe feasibility of distributed algorithms.In this paper, for a given IoT MD i , who is involved in M , we assume it only generates independently offloadabletasks following a Poisson process with an average rate λ i sameas many papers [9], [17], [24]. Each task of IoT MD i ischaracterized by c i and z i , which denote the average numberof CPU cycles required and the average size of computation TABLE I: Notations Notation Description M the number of IoT MDs M the set of IoT MDs N the number of OSPs N e the number of edge computing OSPs N c the number of cloud computing OSPs N the set of OSPs λ i the average task arrival rate of IoT MD ic i the average number of CPU cycles requiredby each task of IoT MD iz i the average size of computation input dataof each task of IoT MD iα i,j the probability that a task is offloaded fromIoT MD i to OSP j α i the offloading strategy vector of IoT MD i A the offloading strategies profile of all IoT MDs A − i the offloading strategies profile of all IoT MDsexcept if MDi the processing capability of IoT MD iε locali the local computing power of IoT MD iε txi the transmission power of IoT MD iB the wireless channel bandwidth w the background interference power h i the channel gain between IoT MD i and BS σ i the variance of service times in the IoT MD i ’s wireless interface a j the number of optical amplifying from theBS to the cloud computation center R the uplink data rate of the optical fiber network t the uplink propagation delay of the opticalbackbone network f OSPj the computing capability of MEC serverowned by OSP jp j the price announced by OSP j p the price vector of all OSPs p − j the prices announced by all OSPs except jD MAXi the maximum computing delay IoT MD i can accept E MAXi the maximum enrgy consumption IoT MD i can accept D MAXi the maximum payment cost IoT MD i can accept θ D i the weight factor of delay for IoT MD iθ E i the weight factor of energy for IoT MD iθ P i the weight factor of payment for IoT MD iU MDi the disutility function of IoT MD iU OSPj the utility function of OSP j input data (e.g., program codes and input parameters) [9]. Thegenerated tasks can not only be executed locally but can alsobe partially offloaded to OSPs’ servers. In other words, IoTMD i can offload each of its tasks to any OSPs. We alsoassume that a task is offloaded from IoT MD i to OSP j witha probability of α i,j (or equivalently the long-term offloadingratio [24]). Therefore, for MD i , there is an offloading strategyvector represented as α i = ( α i, , ..., α i,N ) T . Obviously, vector α i is subjected to the constraint α i,j ∈ [0 , and ≤ (cid:80) j ∈N α i,j ≤ .We summarize all the key notations in Table. I. What needsto be emphasized is, for simplicity, in the rest of the paper,when a task is offloaded to a server belonging to OSP j , wewill directly say that the task is offloaded to OSP j . A. Local Computing
Due to the limited computing capability of an IoT MD,we consider there is an
M/M/ queue model in the localCPU with tasks arriving rate (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i for everyIoT MD i same as [17]. Assuming the processing capabilityof IoT MD i is f MDi , the mean delay caused by computing atask locally in IoT MD i is D locali = c i f MDi − (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i c i . (1)We denote the local computing power of IoT MD i as ε locali ,and the energy consumption of computing a task in IoT MD i is E locali = ε locali c i f MDi − (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i c i . (2)In particularly, (1) and (2) should be subject to the condition (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i c i < f MDi , so as to guarantee queuestability.
B. Computation Offloading
When IoT MD i transmits its tasks to BS, the uplink delayconsists of the waiting time in the wireless interface, theservice time of the wireless interface to submit the tasks to BS.In this paper, we assume that each IoT MD is equipped witha single antenna and the interference cause by other IoT MDscan not be ignored. According to Shannon-Hartley Theorem,the uplink data rate of IoT MD i via wireless cellular networkis r i = B log (cid:32) ε txi h i w + (cid:80) k ∈M ,k (cid:54) = i ε txk h k (cid:33) , (3)where B is wireless channel bandwidth and ε txi is the transmis-sion power of IoT MD i . h i is the channel gain between IoTMD i and BS and w is the background interference power.Obviously, (3) considers the worst case where most channelnoise is brought by all other IoT MDs. It can be noticed thatas the number of IoT MDs in the network increases, the uplinkdata rate of each MD will decrease accordingly.Without loss of generality, we assume that the arrivingtraffic at each IoT MD’s wireless interface, which is the datastream that needs to be uploaded to the BS, is the input datafrom the offloaded tasks, which arrives following a Poissonprocess with arrival rate λ i c i (cid:80) j ∈N α i,j . Thus, the wirelessinterface queue can be modeled as an M/G/ queuing system,and based on the Pollaczed-Khinchin formula [25], the averagewireless transmission delay of IoT MD i can be defined asfollows: D tx,wirelessi = λ i S i (cid:80) j ∈N α i,j (cid:16) − λ i z i (cid:80) j ∈N α i,j r i (cid:17) + z i r i , (4)which is subject to the condition λ i z i (cid:80) j ∈N α i,j < r i . In (4), z i / r i represents the mean service time of each offoaded task inthe wireless interface. S i = σ i + ( z i / r i ) and σ i denotes thevariance of service times. Since the downlink rate is usually much higher than the uplink rate and the size of the outputresults is much smaller than the input size, the download timeof results is neglected as many works [17], [24]. knowing thatthe transmission power of IoT MD i keeps fixed during inputdata transmission, we can get the average energy consumptionfor transmitting a task of IoT MD i to BS as follows: E txi = ε txi D tx,wirelessi = ε txi λ i S i (cid:80) j ∈N α i,j (cid:16) − λ i z i (cid:80) j ∈N α i,j r i (cid:17) + ε txi z i r i . (5)If IoT MD i chooses to offload its tasks to cloud computingOSP j , the input data transmitted to BS should be further de-livered to the cloud computing center via the optical backbonenetwork. To simplify our computation as [8], the total wiredtransmission delay can be denoted as follows: D tx,wiredi,j = (cid:26) a j z i R + t, ≤ j ≤ N c , , otherwise. (6)Here, a j is the number of optical amplifiers from the BSto the cloud computation center owned by OSP j , and R isthe uplink data rate of the optical fiber network. t denotes theuplink propagation delay.In the following, we will study the delay caused by taskcomputation in the server of OSP j . When ≤ j ≤ N c , an M/M/ ∞ queue is modeled for task computation in a cloudcomputation center owing to its rich computing resources.Given that there is only one edge server owned by each edgecomputing OSP, the service queue in an edge computing OSPis modeled as M/M/ . Therefore, we define the averagecomputation delay of a task offloaded from IoT MD i to OSP j as follows: D offloadi,j = (cid:40) c i f OSPj , ≤ j ≤ N c , c i f OSPj − (cid:80) k ∈M α k,j λ k c k , otherwise, (7)where f OSPj is the service rate of a server owned by OSP j . Matrix A − i represents the offloading strategies of all IoTMDs except i . From (7), we can find that when a task of IoTMD i is computed in an edge server, the delay is affected byoffloading strategies of other IoT MDs.Payment cost is also a significant cost in the process oftask offloading. Assuming OSP j charges p j for each unitCPU cycle, when IoT MD i is offloading tasks to OSP j ,the money i should pay to j for making use of computingresources can be denoted as follows: P i,j = p i c i λ i . (8) C. Utility Function
The QoE of an IoT MD can be affected by the comput-ing delay, energy consumption and payment cost. Given theoffloading strategy vector α i , from (1), (2), (4), (5), (6), (7)and (8), we can get the expectation value of computing delay D i ( α i , A − i ) , energy consumption E i ( α i ) and payment cost P i ( α i , p ) for IoT MD i as follows: D i ( α i , A − i ) = (cid:16) − (cid:80) j ∈N α i,j (cid:17) D locali + (cid:80) j ∈N α i,j (cid:16) D tx,wirelessi + D tx,wiredi,j + D offloadi,j (cid:17) , (9) E i ( α i ) = (cid:16) − (cid:88) j ∈N α i,j (cid:17) E locali + (cid:88) j ∈N α i,j E txi , (10) P i ( α i , p ) = (cid:88) j ∈N ( α i,j p j c i λ i ) , (11)where p is the pricing vector of all OSPs.To jointly combine D i ( α i , A − i ) , E i ( α i ) and P i ( α i , p ) asthe quantified form of IoT MD QoE, the weighted method isintroduced to solve this problem as [17]. Different IoT MDshave distinct tolerances of computing delay, energy consump-tion and payment cost, thus we define three constants D MAXi , E MAXi and P MAXi , which represent the maximum computingdelay, energy consumption and payment cost the IoT MD i can accept. Furthermore, for IoT MD i , there are three weightfactors θ D i , θ E i and θ P i , where θ D i + θ E i + θ P i = 1 , reflecting therelative importance of computing delay, energy and paymentfor i . In the distributed task offloading and pricing mechanism,these three weight factors corresponding to an IoT MD i areprivatized by IoT MD i , effectively protecting the privacy ofIoT MDs. From (9), (10), and (11), the disutility function ofIoT MD i can be expressed as follows: U MDi ( α i , A − i , p )= θ D i D i ( α i , A − i ) D MAX + θ E i E i ( α i ) E MAX + θ P i P i ( α i , p ) P MAX . (12)In this paper, we assume that all OSPs do not considerother expenditures apart from the revenue earned by chargingfor the computation offloading. Hence, the utility function ofOSP j can be represented by the revenue function expressedas follows: U OSPj ( p j , p − j , A ) = p j (cid:88) k ∈M λ k c k α k,j , (13)where p − j are the prices announced by all OSPs except j . A is the offloading strategy profile of all MDs. The reason why U OSPj is related to p − j is that the other OSPs’ prices canaffect the offloading strategies of IoT MDs, thereby indirectlyaffecting the revenue of OSP j .IV. G AME ANALYSIS
In this section, first of all, the optimization problems ofeach IoT MD and each OSP are formulated and a multi-leadermulti-follower two-tier Stackelberg game model is applied tostudy the interaction between IoT MDs and OSPs. Then, wedefine the competition among IoT MDs and the competitionamong OSPs as non-cooperative games and prove that bothhave Nash equilibrium solutions, thus proving the existenceof Stackelberg equilibrium (SE).
A. Problem Formulation
To this end, based on previous analytic results on thedisutility function of each IoT MD, the optimization problem of IoT MD i , ∀ i ∈ M , can be formulated as P , which is min α i U MDi ( α i , A − i , p ) s.t. C ≤ (cid:88) j ∈N α i,j ≤ ,C ≤ α i,j ≤ , ∀ j ∈ N ,C (cid:16) − (cid:88) j ∈N α i,j (cid:17) λ i c i < f MDi ,C λ i z i (cid:88) j ∈N α i,j < r i ,C (cid:88) k ∈M ( α k,j λ k c k ) < f OSPj , ∀ j ∈ N ,C D i ( α i , A − i ) ≤ D MAXi ,C E i ( α i ) ≤ E MAXi ,C P i ( α i , p ) ≤ P MAXi . (14)As for OSPs, according to (7), the optimization problem ofOSP j , ∀ j ∈ N , can be formulated as P , which is max p j U OSPj ( p j , p − j , A ) ,s.t. p j ≥ p minj , (15)where p minj is the minimum price OSP j can announce,reflecting OSP j ’s operating cost.From the view of an OSP, having no idea of the utilityfunctions of all IoT MDs, each OSP can only dynamicallyadjust its price according to the amount of tasks offloaded toit. If the price is too high, IoT MDs may offload less tasksto it. Instead, if a low price is announced, a reduced utilitymay appear. As such, a feasible price that can maximize therevenue is pursued by each OSP. B. Multi-leader multi-follower two-tier Stackelberg game
Problems P and P together form a multi-leader multi-follower two-tier Stackelberg game. The leaders of the gameare all OSPs and all MDs are followers. Consequently, theplayer set of the game consists of sets M and N . The objectiveof this game is to find the Stackelberg equilibrium (SE)solution(s) from which neither the leaders nor the followershave incentive to deviate. The formal definition of the SE isgiven as follows. Definition 1. (Stackelberg equilibrium, SE) Let α ∗ i be asolution for P of IoT MD i and p ∗ j be a solution for P of OSP j . Then, the pair ( A ∗ , p ∗ ) is an SE for the proposedStackelberg game if for any A ∈ A and p ∈ P , where A ∆ = ( A i ) i ∈M and P ∆ = ( P i ) i ∈M are respectively thestrategy set of MDs and OSPs, the following conditions aresatisfied: U MDi ( α ∗ i , A ∗− i , p ∗ ) ≤ U MDi ( α i , A ∗− i , p ∗ ) (16) and U OSPj ( p ∗ j , p ∗− j , A ∗ ) ≥ U OSPi ( p j , p ∗− j , A ∗ ) . (17)In our proposed distributed computation offloading andpricing mechanism, we consider both IoT MDs and OSPsto be selfish, which means that IoT MDs compete with eachother for OSPs’ computing resources and each OSP adjust itsprice to improve its revenue in a fully non-cooperated manner. Therefore, the Stackelberg game is composed of a followernon-cooperative game and a leader non-cooperative game. Ifboth the non-cooperative game of followers and leaders aresequentially proven to achieve the Nash equilibrium, an SEsolution surely exists in our proposed Stackelberg game.
C. Follower Non-Cooperative Game Analysis
As followers, given the prices of all OSPs, each IoT MDcompetes with each other to minimize its own disutility.Formally, we define the follower non-cooperative game as G follower = (cid:16) M , {A i } i ∈M , (cid:8) U MDi (cid:9) i ∈M (cid:17) , where M is theset of players. Besides, the payoff profile of G follower consistsof the disutility functions of all IoT MDs as shown by (cid:8) U MDi (cid:9) i ∈M . The Nash equilibrium (NE) of G follower isdefined as follows. Definition 2. (Nash equilibrium (NE) of G follower ) From (14),the Nash equilibrium of game G follower defined above is afeasible offloading strategy profile A ∗ ∆ = ( α ∗ i ) i ∈M , such thateach α ∗ i belongs to A i and satifies the condition as follows: α ∗ i ∈ arg min α i U MDi (cid:0) α i , A ∗− i (cid:1) , (18) which is subject to C - C . Theorem 1.
For ∀ i ∈ M , the set A i is closed and convex andthe function U MDi ( α i , A − i ) is continuously differentiable on A ∆ = ( A i ) i ∈M and convex in α i for every fixed A − i ∈ A − i .Proof. Accoding to [26], a function is convex if and only ifits Hessian matrix is positive semidefinite. Thus, to prove theconvexity of the objective function U MDi ( α i , A − i ) , it sufficesto show that for every fixed A − i ∈ A − i , ∇ α i U MDi (cid:23) . (19)From (12), (19) can be converted to: θ D i ∇ α i D i D MAXi + θ E i ∇ α i E i E MAXi + θ P i ∇ α i P i P MAXi (cid:23) . (20)The (x,y)-th elements of ∇ α i D i , ∇ α i E and ∇ α i P arecalculated as follows: (cid:0) ∇ α i D i (cid:1) xy = ∂ D i ∂α i,x α i,y = (cid:26) Γ i + Υ i + Ψ ix , x = y, N c + 1 ≤ x ≤ N Γ i + Υ i , otherwise, (21) (cid:0) ∇ α i E i (cid:1) xy = ∂ E i ∂α i,x α i,y = ε locali Γ i + ε txi Υ i , (22) (cid:0) ∇ α i P i (cid:1) xy = ∂ P i ∂α i,x ∂α i,y = 0 , (23)where, Γ i = 2 λ i c i (cid:104) f MDi − (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i c i (cid:105) + (cid:16) − (cid:88) j ∈N α i,j (cid:17) λ i c i (cid:104) f MDi − (cid:16) − (cid:80) j ∈N α i,j (cid:17) λ i c i (cid:105) , (24) Υ i = λ i S i (cid:16) − λ i z i (cid:80) j ∈N α i,j r i (cid:17) + (cid:88) j ∈N α i,j z i S i λ i r i (cid:16) − λ i z i (cid:80) j ∈N α i,j r i (cid:17) , (25) Ψ ix = 2 λ i c i (cid:0) f OSPx − (cid:80) k ∈M α k,x λ k c k (cid:1) + α i,x λ i c i (cid:0) f OSPx − (cid:80) k ∈M α k,x λ k c k (cid:1) . (26)From constraints C - C , it is not hard to deduce that Γ i > , Υ i > and Ψ ix > , for ∀ i ∈ M and ∀ x ∈ N .After calculation, we find that ∇ α i D i , ∇ α i E i and ∇ α i P i are all positive (semi)definite matrices. Therefore, we caneasily conclude that (17) is valid.Besides, we can easily find that the set A i is closed for ∀ i ∈ M . To prove the convexity of the set A i , we need toconfirm that the nonlinear constraints of objective functionare convex. The convexity of nonlinear constraints can alsobe proved in the same manner as above. Theorem 2.
The game G follower is equivalent to the varia-tional inequality problem V I ( A , F ) , where F ( α i , A − i ) = ( F i ( α i , A − i )) i ∈M , (27) with F i ( α i , A − i ) = ∇ α i U MDi ( α i , A − i ) . (28) Proof.
According to Proposition 2 in [27], the game G followr can be equivalent to the variational inequality problem V I ( A , F ) , supposing that for each player IoT MD i , thefollowing conditions hold:1) the (nonempty) strategy set A i is closed and convex; the payoff function U MDi ( α i , A − i ) is convex and con-tinuously differentiable in α i for every fixed A − i . From Theorem 2, we can find that the two conditions are bothsatisfied by our formulated follower game G follower . Thus, theresult follows.The reason why we carry out the well-developed VI theoryto analyze follower non-cooperative game is that the prop-erties of V I ( A , F ) can reflect the existence/uniqueness of G follower ’s Nash equilibrium. Next, Theorem 3 is proposedto prove that the G follower has such Nash equilibrium asDefinition 2. Theorem 3.
The mapping F is a monotone function on A , and the game G follower has a convex Nash equilibriumsolution set.Proof. To begin with, we will classify the mapping F as amonotone function on A . According to Definition 40 in [27], F is monotone on A if for all pairs x and y in A , ( x − y ) T ( F ( x ) − F ( y )) ≥ (29)that is M (cid:88) i =1 ( x i − y i ) T ( F i ( x ) − F i ( y )) ≥ . (30) According to [28], if for ∀ i ∈ M , F i meets the conditionas follows: M (cid:88) i =1 ( x i − y i ) T ( F i ( x ) − F i ( y )) ≥ , (31)(28) holds. From Theorem 2, we can see that the Jacobianmatrix J α i F i = ∇ α i U MDi is positive definite and condition(29) holds. Therefore, we can conclude that F is monotoneon A . Next, as described in Theorem 3 of [27], based on themonntonity of F on A , the game G follower has a convex Nashequilibrium solution set. D. Leader Non-Cooperative Game Analysis
As leaders, having no idea of the utility functions of IoTMDs, OSPs can only dynamically adjust their prices accordingto the amount of tasks offloaded to them. The relationshipamong OSPs is competitive with no enforced rules and anOSP has no idea of other OSPs’ pricing strategies. There-fore a leader non-cooperative game for OSPs is defined as G leader = (cid:16) N , {P j } j ∈N , (cid:8) U OSPj (cid:9) j ∈N (cid:17) , where N denotes theset of players and the strategy set of OSP j is all prices abovecost, i.e., P j = (cid:8) p j | p j ≥ p minj (cid:9) . The Nash equilibrium (NE)of G leader is defined as follows. Definition 3. (Nash equilibrium (NE) of G leader ) From (15),the Nash equilibrium of game G leader defined above is afeasible pricing strategy profile p ∗ ∆ = (cid:0) p ∗ j (cid:1) j ∈N , such that each p ∗ j belongs to P j and satifies the conditions as follows: p ∗ j ∈ arg max p j U OSPj (cid:0) p j , p ∗− j (cid:1) ,s.t. p j ≥ p minj . (32)Here, we consider that the offloading strategies of all IoTMDs are only determined by OSPs’ prices. Therefore, theutility of the focused OSP will be limited only by its ownprice if the prices of other OSPs are given. Based on this,the following theorem about the existence of NE of G leader isanalyzed. Theorem 4.
Toward the leader non-cooperative game G leader ,there always exists at least one NE, if for ∀ i ∈ M , thefollowing condition holds: i (cid:34) c i (cid:0) f MDi − λ i c i (cid:1) + λ i c i (cid:0) f MDi − λ i c i (cid:1) (cid:35) ≤ Θ i s i z i r i , (33) where Π i = P MAXi θ P i λ i c i (cid:18) θ D i D MAXi + θ E i ε locali E MAXi (cid:19) , (34) Θ i = P MAXi θ P i λ i c i (cid:18) θ D i D MAXi + θ E i ε txi E MAXi (cid:19) . (35) Proof.
From the analysis of follower non-cooperative game G follower , for a focused OSP j , if the prices of other OSPsare given, we can get the relationship between p j and α i,j from the first-order partial derivative of IoT MD i ’s disutilityfunctions, i.e., p j = RE i ( α i,j ) = p j − P MAXi θ Pi λ i c i · ∂U MDi ∂α i,j , ∀ i ∈ M . (36) Therefore, for each OSP j , adjusting a proper price tomaximize its utility is equivalent to sovling the problem asfollows: max p j p j (cid:88) i ∈M RE − i ( p j ) λ i c i ,s.t. p j ≥ p minj , (37)To prove the existence of NE, we must prove that (37)is concave, in other words, we should prove that the secondderivative of the objective is always not greater than 0, i.e., (cid:88) i ∈M λ i c i (cid:104) (cid:0) RE − i (cid:1) (cid:48) + p j (cid:0) RE − i (cid:1) (cid:48)(cid:48) (cid:105) = (cid:88) i ∈M λ i c i (cid:34) RE i ) (cid:48) + p j − ( RE i ) (cid:48)(cid:48) (cid:0) ( RE i ) (cid:48) (cid:1) (cid:35) ≤ , (38)where ( RE i ) (cid:48) and (cid:0) RE − i (cid:1) (cid:48) are the first derivative functions of RE i ( α i,j ) and its inverse function respectively. ( RE i ) (cid:48)(cid:48) and (cid:0) RE − i (cid:1) (cid:48)(cid:48) are the second derivative functions of RE i ( α i,j ) and its inverse function.After calculation, we can easily find that ( RE i ) (cid:48) is alwaysless than 0. In order to ensure that for ∀ p j ≥ p minj , inequality(38) always holds, ( RE i ) (cid:48)(cid:48) ≤ should always hold. Byanalyzing the analytical formula of ( RE i ) (cid:48)(cid:48) , we find that when α i takes , the value of ( RE i ) (cid:48)(cid:48) is largest. According to max (cid:8) ( RE i ) (cid:48)(cid:48) (cid:9) ≤ , the condition (33) holds. Therefore, theproof of Theorem 4 is proved.V. A LGORITHM D ESIGN
In this section, based on the game theory analysis in Sec.IV, we propose a fully distributed task offloading and pricingmechanism in MEC enabled edge-cloud systems as Fig. 2. Inthe leader non-cooperative game, the OSPs must wait for thefollower non-cooperative game to achieve Nash equilibriumafter announcing their prices. The collection and distributionof OSPs’ prices and IoT MDs’ offloading strategies are alldone through the BS. Some public knowledge includes theaverage task arrival rate and average number of CPU cyclesrequired for each task of each IoT MD is learnt by BS as soonas the IoT MD moves into the coverage of the BS.This section is composed of two subsections. In the firstsubsection, we will deal with the first condition where theprices charged by all OSPs are confirmed, and a distributediterative algorithm is proposed to obtain equilibrious offload-ing strategies for all IoT MDs. In the second subsection, weput forward an algorithm to determine OSPs’ pricing strategiesiteratively and to achieve approximate Stackelberg equilibriumpursuantly. Both of these iterative algorithms ensure littlecommunication overhead thanks to the negligible size of thecommunication message.
A. IPOA
As described in [27], there are a host of solution methodsavailable in the literature to solve monotone real VIs and thusmonotone Nash equilibrium problems, but these algorithmsare centralized and unsuitable in the absence of authority.With regards to this matter, we will introduce an iterativeproximal offloading algorithm (IPOA) to solve the follower
OSPprices offloadingstrategy offloadingrequestspricesOSPT t t+1 OSPsIoT MDsIoT MD Follower non-cooperative gameLeader non-cooperative gameIoT MD IoT MD IoT MD T+1
Fig. 2: Distributed task offloading and pricing mechanismnon-cooperative game G follower that has multiple Nash equi-librium solutions. The iterative algorithm allows all IoT MDsto update their strategies simultaneously in a decentralizedmanner.Before beginning introducing the IPOA in detail, we con-sider a perturbation of game G follower defined as G followerτ, β = (cid:18) M , A , (cid:104) U MDi + ( τ / (cid:107) α i − β i (cid:107) (cid:105) i ∈M (cid:19) , where τ is a pos-itive parameter and β = ( β i ) i ∈M with each β i ∈ R N . TheNash equilibrium of G followerτ, β is each IoT MD i obtain itsoptimal offloading strategy α ∗ i ∈ A i to solve the followingconvex optimization problem: min α i U MDi ( α i , A − i ) + τ (cid:107) α i − β i (cid:107) , (39)is subject to C - C . When τ is large enough, A ∗ is a Nashequilirium solution of G if and only if A ∗ is a Nash equiliriumsolution of G τ, A ∗ . The way to obtain a Nash equiliriumsolution of G τ, A ∗ is described thoroughly in Algorithm 1.At the beginnig, given p = ( p j ) j ∈N , we initialize theoffloading strategies of all IoT MDs to feasible values A (0) ,and set the number of iterations to 0. Moreover, the BS sendsthe service capabilities collected from all OSPs to all IoTMDs. Within iteration l , each IoT MD i deals with its convexoptimization problem (39) with several linear and nonlinearconstraints using Interior Point Method and sends its ownoffloading strategy to BS. After the last IoT MD sending itsnewly updated strategy to the BS, the BS broadcasts A ( l ) to all IoT MDs and informs them to update their centroid( β i ). A new iteration begins unless the offloading strategiesof all IoT MDs are fixed. In this paper, we assume thatthe offloading strategies of all IoT MDs are unchanged if (cid:13)(cid:13) A ( l ) − A ( l − (cid:13)(cid:13) ≤ σ where A ( l ) = (cid:16) α ( l ) i (cid:17) i ∈M and σ is a relativly small constant. As we can see, this algorithmeffectively reduces the number of communications betweenOSPs and IoT MDs and it also guarantees the performanceunder the premise of ensuring IoT MDs’ privacy. Given a τ large enough, the game G followerτ,β can converge in finiteiteration times. Due to space limitations, the detailed proofof the convergence of IPOA can be found in Theorem 16 in[27]. We denote I IP OA as the iteration times of the outerfor loop and the computational complexity of
Interior PointMethod used by each IoT MD in each iteration can beconsidered as O ( I IP M ) [29], where I IP M is a finite integer.Therefore, the total running time of IPOA can be calculatedas O ( I IP OA · I IP M · M ) . Algorithm 1
Iterative proximal offloading algorithm (IPOA)
Input: p , σ , M , (cid:8) f OSPj (cid:9) j ∈N Output: A ∗ Initialization: choose a feasible offloading strategy profile A (0) . Set β i ← α (0) i , ∀ i ∈ M and l ← . The BS sends p and (cid:8) f OSPj (cid:9) j ∈N to all IoT MDs. repeat for each MD i ∈ M do Receives A ( l ) − i from the BS, and uses Interior PointMethod to obtain α ( l +1) i as follows: α ( l +1) i ∈ arg min (cid:110) U MDi (cid:16) α i , A ( l ) − i (cid:17) + τ (cid:107) α i − β i (cid:107) (cid:111) Sends the newly updated offloading strategy α ( l +1) i to the BS. end for if the optimal strategies of all IoT MDs in the l -thiteration have been achieved then Each IoT MD updates its centrods, i.e., β i ← α ( l +1) i , ∀ i ∈ M end if Set l ← l + 1 until (cid:13)(cid:13) A ( l ) − A ( l − (cid:13)(cid:13) ≤ σ B. ISPA
Similar as [21] and [22], backward induction is used tosolve the Stackelberg game and we propose an iterativeStackelberg game pricing algorithm (ISPA). The details ofISPA is described below.We first set a feasible value p (0) j ≥ p minj for the initialprice of each OSP j , j ∈ N and obtain the correspondingoffloading strategies of all IoT MDs through IPOA, then theinitial utilities of all OSPs are confirmed. In each iteration k ,each OSP uses the marginal utility function to adjust its pricebased on IoT MDs offloading strategies, i.e., p ( k +1) j = max p ( k ) j + ∆ j ∂U OSPj (cid:16) p ( k ) j , p ( k ) − j , A ( k ) (cid:17) ∂p ( k ) j , p minj , (40) where ∆ j > represents the positive iteration value of OSP j and the partial derivative of U OSPj to p j can be calculatedby a relatively small value η as: ∂U OSPj ( p ( k ) j ,p ( k ) − j ,A ( k ) ) ∂p ( k ) j ≈ U OSPj (cid:16) ...,p ( k ) j + η,... (cid:17) − U OSPj ( ...,p ( k ) j − η,... )2 η (41)Here, we can find that as OSP j adjusts its price to gainthe partial derivatinve of U OSPj to p j , IoT MDs should re-determine their offloading strategies to the small changes inprice p j via IPOA. In each iteration k , each OSP gets itsupdated price, thereby a new collection of OSPs’ utilities U OSP ( k ) = (cid:0) U OSPj ( k ) (cid:1) j ∈N is obtained. As the iterationincreases, the prices gradually approximate the optimal prices.Taking into account the dynamics of the mobile network,in order to quickly obtain approximate equilibrium pricesof all OSPs, we set I ISP A as the maximum iteration timesof outer for loop. Given the computational complexity ofIPOA is O ( I IP OA · I IP M · M ) , we can get that the totalrunning time of ISPA is O ( I ISP A · I IP OA · I IP M · M · N ) .The pseudocode of the proposed algorithm is presented inAlgorithm 2. Algorithm 2
Iterative Stackelberg game pricing algorithm(ISPA)
Input: ρ , N , M Output: p ∗ , A ∗ Initialization: set a feasible value for p (0) . Get the corre-sponding offloading strategies of all IoT MDs A (0) andcalculate all OSPs’ initial utilities U OSP (0) consequently.Set k ← . repeat for each OSP j ∈ N do Adds an η to its price p ( k ) j and sends it to the BS,then corresponding offloading strategies of all IoTMDs are obtained via IPOA. Reduces an η to its price p ( k ) j and sends it to the BS,then corresponding offloading strategies of all IoTMDs are obtained via IPOA. Updates its price through (40) and (41) and sends theupdated price to the BS. end for if all OSPs have updated their prices then The BS sends p ( k +1) to IoT MDs and get thecorresponding offloading strategies of all IoT MDs A ( k +1) . Calculate U OSP ( k + 1) using p ( k +1) and A ( k +1) . end if Set k ← k + 1 . until k = I ISP A
VI. S
IMULATION R ESULTS
In this section, we conduct extensive simulations to validatethe performance of our proposed two algorithms for MECenabled edge-cloud systems. For experimental purposes, webuild up an IoT scenario consisting of a BS covering a circular TABLE II: Simulation Parameters
Parameter Value Parameter Value λ i [20 , task/min ε txi [0 . , . W c i
300 Mcycles B
100 MHz z i
500 Kb w − Wf MDi [300 , MHz h i -50 dBm ε locali σ i D MAXi f OSPj [1 . , . GHz E MAXi R
10 Gbps P MAXi t area with a radius of 200m. In order to ensure the authenticityand credibility of the parameters, we refer to many papersand standards [9], [17], [30], [31] to design our experimentalsimulation parameters. The detailed parameter settings areshown in Table. II. In Table. II, f MDi and f OSPj take thevalue uniformly in the corresponding value intervals and otherparameters are designated fixed values. For ∀ i ∈ M , theweight factors, θ D i , θ E i and θ P i are randomly retrieved valuesfrom the range [0 , so that they are individually different. A. Task offloading strategies of IoT MDs
In this subsection, we will investigate the convergenceand effectiveness of IPOA in an MEC enabled edge-cloudsystem that includes one cloud computing OSP and three edgecomputing OSPs. Given that the price charged by each cloudcomputing OSP is 0.2 $/Gcycles and that the price of eachedge computing OSP is 0.1 $/Gcycles, we first assume thatthere are 50 IoT MDs within the coverage of the BS and theaverage task arrival rate of each IoT MD is 25 task/min. Inaddition, .As shown in Fig. 3, we select 5 IoT MDs (MD 5, 15,25, 35 and 45), to show the disutility function values of theseIoT MDs versus the number of iterations. From Fig. 3, wecan observe that in the previous iterations, the change in thedisutility function value of each IoT MD fluctuates drasticallyand reaches a relatively stable state in about 7 iterations. Inother words, all IoT MDs achieve Nash equilibrium after alimited number of iterations given the prices of all OSPs.Moreover, considering that the time spent in each iteration isfar less than the task computation time, the IPOA can convergeto a Nash equilibrium very quickly and the high efficiency ofIPOA is thus proved.In Fig. 4 and Fig. 5, we compare the average utility ofall IoT MDs obtained through IPOA against the followingbaselines. • Local Computing : Each IoT MD processes all of its taskslocally. • Cloud Computing : Each IoT MD offloads all of its tasksto cloud computing servers. • Evenly Offloading : Each IoT MD processes a task locallyor each OSP with the same probability. • Socially Optimal Offloading : According to [17], giventhe prices of all OSPs, we can get the socially optimaloffloading strategies of all IoT MDs via an IPM-basedalgorithm.In this paper, we define the price of anarchy (PoA), whichreflects how far is the overall performance of an NE from thesocially optimal offloading scheme, is the average disutility Iterations D i s u tilit y MD 5MD 15MD 25MD 35MD 45
Fig. 3: The disutility versus the iterations of IPOAvalue obtained by IPOA divided by the socially optimalaverage disutility value, i.e.,
P oA = ¯ U MD ( A ∗ )¯ U MD ( A SO ) , (42)where ¯ U MD ( A ∗ ) is IoT MDs’ average disutility obtainedby IPOA and ¯ U MD ( A SO ) is the socially optimal averagedisutility of all IoT MDs. It is not difficult to see that thesmaller the PoA, the closer the performance of IPOA is to thesocially optimal performance.As shown by Fig. 4, when we set the transmission powerof each IoT MD to 400 mW, the IoT MDs’ average disutilityincreases as the average task arrival rate of each IoT MDvaries from 20 task/min to 29 task/min. Additionally, we canobserve that the performance of IPOA is significantly superiorto that of local computing and cloud computing. Although theaverage disutility obtained by IPOA is slightly less than evenlyoffloading, the system performance of IPOA evidently excelsthat of evenly offloading when considering the quantity baseof IoT MDs. Based on (42), the value of PoA decreases from43% to 36% as the average task arrival rate increases, whichmeans the PoA of IPOA is bounded in the real scenes.Fig. 5 demonstrates the impact of ε txi on the averagedisutility of all IoT MDs with the average task arrival rate ofeach IoT MD is set to a fixed value 25 task/min. Obviously,the average disutility caused by local computing is not alteredby ε txi . On the contrary, the increase of ε txi will bring morediutility for all IoT MDs when they offload their tasks throughother offloading strategies. Fig. 5 proves the high performanceof IPOA compared with other baselines and once again verifiesthe NE obtained by IPOA does not bring immeasurableperformance loss to the system. B. Pricing strategies of OSPs
In this subsection, we focus on the OSPs and study theperformance of ISPA. Moreover, the influence of the IoT
20 21 22 23 24 25 26 27 28 29 i (task/min) A v e r age d i s u t ili t y
135 %136 %137 %138 %139 %140 %141 %142 %143 %144 % P r i c e o f ana r c h y Local ComputingCloud ComputingEvenly OffloadingSocially Optimal OffloadingIPOAPoA
Fig. 4: The average disutility versus the average task arrivalrate of IoT MDs itx (W) A v e r age d i s u t ili t y
132 %134 %136 %138 %140 %142 %144 %146 %148 % P r i c e o f ana r c h y Local ComputingCloud ComputingEvenly OffloadingSocially Optimal OffloadingIPOAPoA
Fig. 5: The average disutility versus the transmission powerof IoT MDsMD number on the performance is analyzed in the followingexperiment. In particular, we set the cost price of each OSPas its initial price at the beginning of ISPA.In the case of setting λ i to 25 task/min and ε txi to 400 mW,Fig. 6 presents the changing prices of cloud computing OSPsand edge computing OSPs with the iterations. Specifically, theprice variations corresponding to 5 different IoT MD numbers,10, 30, 50, 70 and 90 respectively, are shown in differentline types. The prices obviously show a rapidly increasingtrend with the iteration increasing during the whole stage.The reason for the obvious fluctuation of the curve instead ofmonotonically increasing with the iterations is that we apply(41) to approximate the derivative direction of each OSP utilityfunction in each iteration. Furthermore, we can see that the Iterations P r i ce ( $ / G c y c l e s ) -10 M = 10, cloud computing OSPM = 10, edge computing OSPM = 30, cloud computing OSPM = 30, edge computing OSPM = 50, cloud computing OSPM = 50, edge computing OSPM = 70, cloud computing OSPM = 70, edge computing OSPM = 90, cloud computing OSPM = 90, edge computing OSP
Fig. 6: Prices of OSPs versus the iterations of ISPAprices of OSPs will increase significantly when the number ofIoT MDs increases and the pricing of edge computing OSP ishigher than that of cloud computing OSP as edge computingOSPs have the advantage of the cloud computing OSPs in theleader non-cooperative game.Under the same simulation settings as in Fig. 6, Fig. 7illustrates the utility changes of the cloud computing OSPsand edge computing OSPs as the iterations progress withvarious IoT MD numbers. We can easily find that the utilitiesof all OSPs increase with each iteration. Besides, each edgecomputing OSP’s utility is always higher than that of the cloudcomputing OSP.In Fig. 8, we compare ISPA with blind pricing with 50IoT MDs. Here, blind pricing is a pricing scheme where eachOSP increases its price without considering the impact ofother OSPs’ pricing. In our simulation, we assume that allblinding OSPs set their cost prices as their initial prices andlinearly increase their prices to the average price of ISPA atthe 50th iteration. As we can see, the average utility of OSPsin ISPA performs better than that of blinding OSPs after the37th iteration and the performance gap between ISPA andblind pricing gradually widens. In the 50th iteration, ISPA canimprove the average utility of OSPs by up to 7.5% comparedwith the bliding pricing scheme at the 50th iteration and agreater improvement can be foreseen if the iteration continues.VII. C
ONCLUSION
This paper provides a novel sdistributed mechanism toaddress the pricing and task offloading in MEC enabled edge-cloud systems. First, we quantify the QoE of IoT MDs with adisutility function which jointly considers the computing delay,energy consumption and payment cost. Then a multi-leadermulti-follower two-tier Stackelberg game model is applied todescribe the optimization problems of OSPs and IoT MDsand the existence of SE is analyzed. We first propose IPOAto obtain Nash equilibrium offloading strategies for IoT MDs
Iterations U tilit y ( $ ) M = 10, cloud computing OSPM = 10, edge computing OSPM = 30, cloud computing OSPM = 30, edge computing OSPM = 50, cloud computing OSPM = 50, edge computing OSPM = 70, cloud computing OSPM = 70, edge computing OSPM = 90, cloud computing OSPM = 90, edge computing OSP
Fig. 7: Utilites of OSPs versus the iterations of ISPA
Iterations P r i ce ( $ / G c y c l e s ) Cloud computing OSP in ISPAEdge computing OSP in ISPABlind pricing (a) Prices change
Iterations A v e r a g e u tilit y ( $ ) ISPABlind pricing (b) Utilities change
Fig. 8: Prices and average utility versus the iterationsin the condition that the prices charged by all OSPs arefixed. Furthermore, considering the privacy of IoT MD utilityinformation and the non-cooperative nature of OSPs, we applybackward induction and introduce ISPA so that OSPs can ad-just their prices dynamically. Through numerical experiments,results show that our proposed mechanism provides a superiorsolution to the optimal pricing and task offloading problem inthe competitive IoT environment.A
CKNOWLEDGMENT
This work was supported in part by National NaturalScience Foundations of China (61821001), YangFan Innova-tive & Entrepreneurial Research Team Project of GuangdongProvince, Fundamental Research Funds for the Central Uni-versities, and Director Foundation of Beijing Key Laboratoryof Work Safety Intelligent Monitoring.R
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